Spatial Characterization Of Two-Photon States - GAP-Optique

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A. The matrix form of the mode function writes T r[ρ 2 q] = dqsdΩsdqidΩidq ′ sdΩ ′ sdq ′ idΩ ′ i × Φ(qs, Ωs, qi, Ωi)Φ ∗ (q ′ s, Ωs, q ′ i, Ωi) × Φ(q ′ s, Ω ′ s, q ′ i, Ω ′ i)Φ ∗ (qs, Ω ′ s, qi, Ω ′ i). (A.7) where the dimension increases as new primed variables appear. Using the matrix notation the integrand becomes Φ(qs, Ωs, qi, Ωi)Φ ∗ (q ′ s, Ωs, q ′ i, Ωi)Φ(q ′ s, Ω ′ s, q ′ i, Ω ′ i)Φ ∗ (qs, Ω ′ s, qi, Ω ′ i) = N 4 exp − 1 2 Xt BX , (A.8) where the vector X is the result of concatenation of x and x ′ , such that ⎛ ⎜ X = ⎜ ⎝ ↑ x ↓ ↑ x ′ ↓ ⎞ ⎟ , ⎟ ⎠ (A.9) and the new matrix B is given by B = 1 ⎛ ⎜ 2 ⎜ ⎝ 2a 2h 2i 2j k l 0 0 0 0 k 2h 2b 2m 2n p r 0 0 0 0 p 2i 2m 2c 2s t u 0 0 0 0 t 2j 2n 2s 2d v w 0 0 0 0 v k p t v 2f 2z k p t v 0 l r u w 2z 2g l r u w 0 0 0 0 0 k l 2a 2h 2i 2j k 0 0 0 0 p r 2h 2b 2m 2n p 0 0 0 0 t u 2i 2m 2c 2s t 0 0 0 0 v w 2j 2n 2s 2d v k p t v 0 0 k p t v 2f l r u w 0 0 l r u w 2z ⎞ ⎟ . ⎟ ⎠ l r u w 0 0 l r u w 2z 2g (A.10) In an analogous way, the integrand on the expression for the signal photon purity T r[ρ 2 signal] = is written in a matrix notation as 66 dqsdΩsdqidΩidq ′ sdΩ ′ sdq ′ idΩ ′ i × Φ(qs, Ωs, qi, Ωi)Φ ∗ (q ′ s, Ω ′ s, qi, Ωi) × Φ(q ′ s, Ω ′ s, q ′ i, Ω ′ i)Φ ∗ (qs, Ωs, q ′ i, Ω ′ i). (A.11) Φ(qs, Ωs, qi, Ωi)Φ ∗ (q ′ s, Ω ′ s, qi, Ωi)Φ(q ′ s, Ω ′ s, q ′ i, Ω ′ i)Φ ∗ (qs, Ωs, q ′ i, Ω ′ i) = N 4 exp − 1 2 Xt CX . (A.12)
The matrix C is given by C = 1 ⎛ 2a ⎜ 2h ⎜ i ⎜ j ⎜ 2k ⎜ l 2 ⎜ 0 ⎜ 0 ⎜ i ⎜ j ⎝ 0 2h 2b m n 2p r 0 0 m n 0 i m 2c 2s t 2u i m 0 0 t j n 2s 2d v 2w j n 0 0 v 2k 2p t v 2f z 0 0 t v 0 l r 2u 2w z 2g l r 0 0 z 0 0 i j 0 l 2a 2h i j 2k 0 0 m n 0 r 2h 2b m n 2p i m 0 0 t 0 i m 2c 2s t j n 0 0 v 0 j n 2s 2d v 0 0 t v 0 z 2k 2p t v 2f l r 0 0 z 0 l r 2u 2w z l r 0 0 z 0 l r 2u 2w z 2g ⎞ ⎟ . ⎟ ⎠ (A.13) 67
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DOCTORAL THESIS IN PHOTONICS ICFO B
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Spatial Characterization Of Two-Pho
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Contents Contents vii Acknowledgeme
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Acknowledgements This thesis compil
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Abstract In the same way that elect
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Resumen De la misma manera que la e
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Introduction The role of photons in
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This thesis is based on the followi
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1. General description of two-photo
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CHAPTER 2 Correlations and entangle
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2.1. The purity as a correlation in
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2.2. Correlations between space and
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2.2. Correlations between space and
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2.3. Correlations between signal an
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2.3. Correlations between signal an
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Signal purity 1 0 (a) 0.1 3 2.3. Co
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CHAPTER 3 Spatial correlations and
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3.2. OAM transfer in general SPDC c
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3.2. OAM transfer in general SPDC c
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Phase front 3.3. OAM transfer in co
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4. OAM transfer in noncollinear con
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5. Spatial correlations in Raman tr
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CHAPTER 6 Summary This thesis chara
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APPENDIX A The matrix form of the m
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v =γ 2 L 2 Np sin ϕi − γ 2 L 2
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The matrix C is given by C = 1 ⎛
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B. Integrals of the matrix mode fun
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C. Methods for OAM measurements Inp
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Bibliography [13] M. Barbieri, C. C
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Bibliography [41] C. I. Osorio, A.
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Bibliography [68] S. Chen, Y.-A. Ch
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Spatial Characterization Of Two-Pho
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A Luz Stella y Luis Alfonso, mis pa
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Contents B Integrals of the matrix
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When he [Kepler] found that his lon
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Abstract The matrix notation, intro
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Abstract La notación matricial int
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Introduction the transfer of oam fr
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CHAPTER 1 General description of Tw
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y y x z z pump s i (a) signal idle
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1.3. Approximations and other consi
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x 1.3. Approximations and other con
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x Wave fronts 1.3. Approximations a
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1 0 -0.2 1.3. Approximations and ot
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1.4. The mode function in matrix fo
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Page 163 and 164: Coincidences 800 0 -4 4.2. Effect o
Page 165 and 166: Weight 1 0 4.3. Effect of the Poynt
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Page 169 and 170: Before the crystal 4.3. Effect of t
Page 171 and 172: CHAPTER 5 Spatial correlations in R
Page 173 and 174: x y z 5.1. The quantum state of Sto
Page 175 and 176: where 5.2. Orbital angular momentum
Page 177 and 178: weight 1 0.6 -180º -90º 0º 90º
Page 179: 5.3. Spatial entanglement 5.20 is s
Page 182 and 183: 6. Summary Experiments that explain
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Page 189 and 190: APPENDIX B Integrals of the matrix
Page 191 and 192: APPENDIX C Methods for OAM measurem
Page 193 and 194: Bibliography [1] M. A. Nielsen and
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Page 197 and 198: Bibliography [55] M. C. Booth, M. A
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Spatial Characterization Of Two-Photon States - GAP-Optique

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